doi: 10.3934/cpaa.2021173
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Marcinkiewicz integral operators along twisted surfaces

1. 

Department of Mathematics, Sultan Qaboos University, Sultanate of Oman

2. 

Department of Mthematics, Yarmouk University, Irbid, Jordan

* Corresponding author

Received  June 2021 Revised  September 2021 Early access September 2021

Marcinkiewicz integral operators on product domains defined by translates determined by twisted surfaces are introduced. Maximal functions along twisted surfaces are also introduced. Conditions on the underlined surfaces implying that the corresponding Marcinkiewicz integral operators map $ L^{p}\rightarrow L^{p} $ for $ 1<p<\infty $ are obtained. A general method involving lacunary families of multi-indices is developed.

Citation: Ahmad Al-Salman. Marcinkiewicz integral operators along twisted surfaces. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021173
References:
[1]

H. Al-QassemA. Al-SalmanL. Cheng and Y. Pan, Marcinkiewicz integrals on product spaces, Stud. Math., 167 (2005), 227-234.  doi: 10.4064/sm167-3-4.  Google Scholar

[2]

H. Al-Qassem and A. Al-Salman, Rough Marcinkiewicz integral operators, Int. J. Math. Math. Sci., 27 (2001), 495-503.  doi: 10.1155/S0161171201006548.  Google Scholar

[3]

A. Al-SalmanH. Al-Qassem and Y. Pan, Singular integrals on product domains, Indiana Univ. Math. J., 55 (2006), 369-387.  doi: 10.1512/iumj.2006.55.2626.  Google Scholar

[4]

A. Al-Salman, Singular integral operators on product domains along twisted surfaces, Front. Math. China, 16 (2021), 13-28.  doi: 10.1007/s11464-021-0911-z.  Google Scholar

[5]

A. Al-Salman, Maximal functions along surfaces on product domains, Anal. Math., 34 (2008), 163-175.  doi: 10.1007/s10476-008-0301-8.  Google Scholar

[6]

A. Al-Salman, Rough Marcinkiewicz integrals on product spaces, Int. Math. Forum, 2 (2007), 1119-1128.  doi: 10.12988/imf.2007.07097.  Google Scholar

[7]

A. Al-Salman, Maximal operators with rough kernels on product domains, J. Math. Anal. Appl., 311 (2005), 338-351.  doi: 10.1016/j.jmaa.2005.02.048.  Google Scholar

[8]

A. Al-Salman and H. Al-Qassem, Rough singular integrals on product spaces, Int. J. Math. Math. Sci., 67 (2004), 3671-3684.  doi: 10.1155/S0161171204312342.  Google Scholar

[9]

A. Al-Salman and Y. Pan, Singular integrals with rough kernels in$Llog^{+}L(S^{n-1})$, J. London Math. Soc., 66 (2002), 153-174.  doi: 10.1112/S0024610702003241.  Google Scholar

[10]

A. Al-SalmanH. Al-QassemL. Cheng and Y. Pan, $L^{p}$ boundes for the functions of Marcinkiewicz, Math. Res. Lett., 9 (2002), 697-700.  doi: 10.4310/MRL.2002.v9.n5.a11.  Google Scholar

[11]

A. Al-Salman and H. Al-Qassem, Integral operators of Marcinkiewicz type, J. Integral Equ. Appl., 14 (2002), 343-354.  doi: 10.1216/jiea/1181074927.  Google Scholar

[12]

A. BenedekA. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci., 48 (1962), 356-365.  doi: 10.1073/pnas.48.3.356.  Google Scholar

[13]

J. ChenD. Fan and Y. Ying, Rough Marcinkiewicz integrals with $L(\log L)^{2}$kernels, Adv. Math. (China), 30 (2001), 179-181.   Google Scholar

[14]

Y. Choi, Marcinkiewicz integrals with rough homogeneous kernel of degree zero in product domains, J. Math. Appl., 261 (2001), 53-60.  doi: 10.1006/jmaa.2001.7465.  Google Scholar

[15]

J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Ins. Fourier (Grenoble), 36 (1986), 185-206.   Google Scholar

[16]

J. Duoandikoetxea and J. L. Rubio de Francia, Maximal functions and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541-561.  doi: 10.1007/BF01388746.  Google Scholar

[17]

Y. Ding, L$^{2}$-boundedness of Marcinkiewicz integral with rough kernel, Hokkaido Math. J., 27 (1998), 105-115.  doi: 10.14492/hokmj/1351001253.  Google Scholar

[18]

D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119 (1997), 799-839.   Google Scholar

[19]

R. Feferman and E. M. Stein, Singular integrals on product spaces, Adv. Math., 45 (1982), 117-143.  doi: 10.1016/S0001-8708(82)80001-7.  Google Scholar

[20]

F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals, J. Funct. Anal., 73 (1987), 179-194.  doi: 10.1016/0022-1236(87)90064-4.  Google Scholar

[21]

E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proc. Int l. Cong. Math., (1986), 196–221.  Google Scholar

[22]

E. M. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430-466.  doi: 10.2307/1993226.  Google Scholar

[23] E. M. Stein, Harmonic Analysis: Real-Variable Mathods, Orthogonality and Oscillatory integrals, Princeton University Press, Princeton, NJ, 1993.   Google Scholar
[24]

H. Wu, Boundedness of multiple Marcinkiewicz integral operators with rough kernels, J. Korean Math. Soc., 43 (2006), 635-658.  doi: 10.4134/JKMS.2006.43.3.635.  Google Scholar

show all references

References:
[1]

H. Al-QassemA. Al-SalmanL. Cheng and Y. Pan, Marcinkiewicz integrals on product spaces, Stud. Math., 167 (2005), 227-234.  doi: 10.4064/sm167-3-4.  Google Scholar

[2]

H. Al-Qassem and A. Al-Salman, Rough Marcinkiewicz integral operators, Int. J. Math. Math. Sci., 27 (2001), 495-503.  doi: 10.1155/S0161171201006548.  Google Scholar

[3]

A. Al-SalmanH. Al-Qassem and Y. Pan, Singular integrals on product domains, Indiana Univ. Math. J., 55 (2006), 369-387.  doi: 10.1512/iumj.2006.55.2626.  Google Scholar

[4]

A. Al-Salman, Singular integral operators on product domains along twisted surfaces, Front. Math. China, 16 (2021), 13-28.  doi: 10.1007/s11464-021-0911-z.  Google Scholar

[5]

A. Al-Salman, Maximal functions along surfaces on product domains, Anal. Math., 34 (2008), 163-175.  doi: 10.1007/s10476-008-0301-8.  Google Scholar

[6]

A. Al-Salman, Rough Marcinkiewicz integrals on product spaces, Int. Math. Forum, 2 (2007), 1119-1128.  doi: 10.12988/imf.2007.07097.  Google Scholar

[7]

A. Al-Salman, Maximal operators with rough kernels on product domains, J. Math. Anal. Appl., 311 (2005), 338-351.  doi: 10.1016/j.jmaa.2005.02.048.  Google Scholar

[8]

A. Al-Salman and H. Al-Qassem, Rough singular integrals on product spaces, Int. J. Math. Math. Sci., 67 (2004), 3671-3684.  doi: 10.1155/S0161171204312342.  Google Scholar

[9]

A. Al-Salman and Y. Pan, Singular integrals with rough kernels in$Llog^{+}L(S^{n-1})$, J. London Math. Soc., 66 (2002), 153-174.  doi: 10.1112/S0024610702003241.  Google Scholar

[10]

A. Al-SalmanH. Al-QassemL. Cheng and Y. Pan, $L^{p}$ boundes for the functions of Marcinkiewicz, Math. Res. Lett., 9 (2002), 697-700.  doi: 10.4310/MRL.2002.v9.n5.a11.  Google Scholar

[11]

A. Al-Salman and H. Al-Qassem, Integral operators of Marcinkiewicz type, J. Integral Equ. Appl., 14 (2002), 343-354.  doi: 10.1216/jiea/1181074927.  Google Scholar

[12]

A. BenedekA. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci., 48 (1962), 356-365.  doi: 10.1073/pnas.48.3.356.  Google Scholar

[13]

J. ChenD. Fan and Y. Ying, Rough Marcinkiewicz integrals with $L(\log L)^{2}$kernels, Adv. Math. (China), 30 (2001), 179-181.   Google Scholar

[14]

Y. Choi, Marcinkiewicz integrals with rough homogeneous kernel of degree zero in product domains, J. Math. Appl., 261 (2001), 53-60.  doi: 10.1006/jmaa.2001.7465.  Google Scholar

[15]

J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Ins. Fourier (Grenoble), 36 (1986), 185-206.   Google Scholar

[16]

J. Duoandikoetxea and J. L. Rubio de Francia, Maximal functions and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541-561.  doi: 10.1007/BF01388746.  Google Scholar

[17]

Y. Ding, L$^{2}$-boundedness of Marcinkiewicz integral with rough kernel, Hokkaido Math. J., 27 (1998), 105-115.  doi: 10.14492/hokmj/1351001253.  Google Scholar

[18]

D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119 (1997), 799-839.   Google Scholar

[19]

R. Feferman and E. M. Stein, Singular integrals on product spaces, Adv. Math., 45 (1982), 117-143.  doi: 10.1016/S0001-8708(82)80001-7.  Google Scholar

[20]

F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals, J. Funct. Anal., 73 (1987), 179-194.  doi: 10.1016/0022-1236(87)90064-4.  Google Scholar

[21]

E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proc. Int l. Cong. Math., (1986), 196–221.  Google Scholar

[22]

E. M. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430-466.  doi: 10.2307/1993226.  Google Scholar

[23] E. M. Stein, Harmonic Analysis: Real-Variable Mathods, Orthogonality and Oscillatory integrals, Princeton University Press, Princeton, NJ, 1993.   Google Scholar
[24]

H. Wu, Boundedness of multiple Marcinkiewicz integral operators with rough kernels, J. Korean Math. Soc., 43 (2006), 635-658.  doi: 10.4134/JKMS.2006.43.3.635.  Google Scholar

[1]

Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915

[2]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7

[3]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1

[4]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027

[5]

Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005

[6]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579

[7]

Zhenbing Gong, Yanping Chen, Wenyu Tao. Jump and variational inequalities for averaging operators with variable kernels. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1851-1866. doi: 10.3934/cpaa.2021045

[8]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[9]

Michael Music. The nonlinear Fourier transform for two-dimensional subcritical potentials. Inverse Problems & Imaging, 2014, 8 (4) : 1151-1167. doi: 10.3934/ipi.2014.8.1151

[10]

Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339

[11]

Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems & Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885

[12]

Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007

[13]

Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020

[14]

Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834

[15]

Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249

[16]

Jae Gil Choi, David Skoug. Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3829-3842. doi: 10.3934/cpaa.2020169

[17]

Nam Yul Yu. A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences. Advances in Mathematics of Communications, 2014, 8 (2) : 209-222. doi: 10.3934/amc.2014.8.209

[18]

Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401

[19]

Patricio Felmer, Alexander Quaas. Fundamental solutions for a class of Isaacs integral operators. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 493-508. doi: 10.3934/dcds.2011.30.493

[20]

Fuzhi Li, Yangrong Li, Renhai Wang. Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3663-3685. doi: 10.3934/dcds.2018158

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (53)
  • HTML views (57)
  • Cited by (0)

Other articles
by authors

[Back to Top]